Limits and Continuity

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Limits and Continuity

• Definition

• Evaluation of Limits

• Continuity

• Limits Involving Infinity

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Limit
L
a
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Limits, Graphs, and Calculators
Graph 1
Graph 2
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Graph 3
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c) Find
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Note f (-2) 1 is not involved

• 2

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3) Use your calculator to evaluate the
limits
Answer 16
Answer no limit
Answer no limit
Answer 1/2
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The Definition of Limit
L
a
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Examples
What do we do with the x?
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1/2
1
3/2
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One-Sided Limits
One-Sided Limit
The right-hand limit of f (x), as x approaches a,
equals L
written
if we can make the value f (x) arbitrarily close
to L by taking x to be sufficiently close to the
right of a.
L
a
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The left-hand limit of f (x), as x approaches a,
equals M
written
if we can make the value f (x) arbitrarily close
to L by taking x to be sufficiently close to the
left of a.
M
a
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Examples of One-Sided Limit
Examples
1. Given
Find
Find
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More Examples
Find the limits
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A Theorem
This theorem is used to show a limit does not
exist.
For the function
But
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Limit Theorems
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Examples Using Limit Rule
Ex.
Ex.
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More Examples
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Indeterminate Forms
Indeterminate forms occur when substitution in
the limit results in 0/0. In such cases either
factor or rationalize the expressions.
Notice form
Ex.
Factor and cancel common factors
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More Examples
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The Squeezing Theorem
See Graph
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Continuity

A function f is continuous at the point x a if
the following are true
f(a)
a
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A function f is continuous at the point x a if
the following are true
f(a)
a
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Examples
At which value(s) of x is the given function
discontinuous?
Continuous everywhere
Continuous everywhere except at
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and
and
Thus F is not cont. at
Thus h is not cont. at x1.
F is continuous everywhere else
h is continuous everywhere else
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Continuous Functions
If f and g are continuous at x a, then
A polynomial function y P(x) is continuous at
every point x.
A rational function is
continuous at every point x in its domain.
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Intermediate Value Theorem
If f is a continuous function on a closed
interval a, b and L is any number between f (a)
and f (b), then there is at least one number c in
a, b such that f(c) L.
f (b)
L
f (c)
f (a)
a
b
c
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Example
f (x) is continuous (polynomial) and since f (1)
lt 0 and f (2) gt 0, by the Intermediate Value
Theorem there exists a c on 1, 2 such that f
(c) 0.
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Limits at Infinity
For all n gt 0,
provided that is defined.
Divide by
Ex.
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More Examples
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Infinite Limits
For all n gt 0,
More Graphs
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Examples
Find the limits
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Limit and Trig Functions
From the graph of trigs functions
we conclude that they are continuous everywhere
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Tangent and Secant
Tangent and secant are continuous everywhere in
their domain, which is the set of all real
numbers
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Examples
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Limit and Exponential Functions
The above graph confirm that exponential
functions are continuous everywhere.
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Asymptotes
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Examples
Find the asymptotes of the graphs of the functions
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